Analyse my game

I played an interesting game today. I was wondering if the stronger players in the community can tell me where I went wrong, what I should work on, any mistakes in my analysis and how I should try to play these sort of positions. Thanks. lichess.org/study/O8ys6R2K

7 Be2 is a well respected line.
9 o-o is good. It is not true that o-o-o is better. It is sharper. White has a faster and more virulent attack, but black gets a more dangerous counterattack. With o-o the White King is safer.
13...Rxc3 is bad. 13...Bc6 or 13...Qa5 are right. If he takes back 14 Qxc3 Nxe4 15 Qd3 he is material up and has no weak pawn on c3.
15...Bf5 is wrong. Just strike at the weak pawn with 15...Qc8.
29...Kg8 is a blunder. Just 29...e5 could have held. You lose the rook on b2, but you have 5 pawns for it and his King is in the open. It ends probably in a perpetual check.

Thanks so much @tpr

Are 5 pawns really worth a rook?

Generally speaking 5 pawns are about worth a rook, but it depends on the concrete position. Here the problem is not that black's pawns are going to promote to queen, but that the white King has no pawns to shelter him, and thus black can give perpetual check with his queen despite being a rook down.

Ok thanks

If the queens are exchanged 5 pawns might even be winning against the rook so white has to probably accept a perpetual.

White cannot avoid the perpetual as he cannot shelter his King. Black can choose not to give perpetual, but the pawns are not sufficiently advanced and then the rook wil prevail.

ya

For me one point is looking at tactical oversights. Would you have played Rxc3 in case you had seen QxRc3? If yes you need to play more of these positions for better judgement.

Similar is Kg8 vs. e5 at the end of the game.